Apparatus and method for outputting sequence of vectors, data recording medium, and carrier wave signal

ABSTRACT

A first storage in an apparatus for outputting a sequence of vectors stores vector x having the dimension at least 1, a first calculator calculates vector function x′=f(x) which utilizes a first rational vector map f to which the stored vector x is input, a second storage stores vector y having the dimension at least 1, a second calculator calculates vector function y′=g(x, y) which utilizes a second rational vector map g to which the stored vectors x and y are input, an output outputs vector z′ which is association of the resultant vectors x′ and y′, a first update updates the first storage by storing the resultant vector x′, and a second update updates the second storage by storing the resultant vector y′.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The present invention relates to an apparatus and method for outputtinga sequence of vectors, a data recording medium, and a carrier wavesignal.

More particularly, the present invention relates to an apparatus andmethod for outputting a sequence of higher dimensional random vectorswhose limiting distribution is expressed by an analytical densityfunction, after associating two vector sequence generating methods bywhich sequences of random vectors whose limiting distribution isexpressed by a known analytical density function are output, and a datarecording medium storing a program which realize the above vectorsequence output.

2. Description of the Related Art

Many random number generating methods utilizing recurrence formulas havebeen known conventionally. Many fields require random number generation.Monte-Carlo method for simulation in physics and engineering fieldsutilizes the random numbers.

CDMA (Code Division Multiple Access) technology for mobile phonecommunication assigns PN (Pseudo Noise) code to each user in order toshare the limited band among many users. The PN code is generated basedon the random numbers.

Moreover, public key encryption employed in telecommunicationtechnologies utilizes the random numbers for generating public keys.Demands for such the encryption has been developing because strongerprotection has been required as telecommunications such as internet hasbeen widely used.

Traditional methods for generating the random numbers usually utilizerecurrence formulas. Especially, multiplication recurrence formulas havebeen used widely for many years. However, such the multiplicationrecurrence formula also has raised problems regarding to finiteperiodicity. Recently, rational maps have been applied to the recurrenceformulas to generate random numbers as chaos theory has been developing.The rational map is a result of the addition theorem of an ellipticfunction (including a trigonometric function). Demands for the randomnumber generation by such the method which utilizes the rational maphave been developing, because such the method has the followingadvantages.

(1) Non-cyclical random number sequence generation by which the numbersare proven to be chaotic (thus, aperiodic):

(2) Sequences of rational numbers result from rational number seeds(initial values given to the recurrence formula); and

(3) Known analytic function acts as the density function expressingrandom number distribution.

Known rational maps which bring the above advantages are: an Ulam-vonNeumann map (equation 1), a cubic map (equation 2), a quintic map(equation 3), and the like. $\begin{matrix}{{f(x)} = {4{x\left( {1 - x} \right)}}} & {{EQUATION}\quad 1} \\{{f(x)} = {x\left( {3 - {4x}} \right)}^{2}} & {{EQUATION}{\quad \quad}2} \\{{f(x)} = {x\left( {5 - {20x} + {16x^{2}}} \right)}^{2}} & {{EQUATION}{\quad \quad}3}\end{matrix}$

Regardless of the different rational maps above, the equation 4represents a density function expressing distribution of a random numbersequence x[i] which results from the following recurrence formula (where0<ξ<1; ξ is an arbitrary initial value).

 x[0]=ξ

x[i+1]=f(x[i]) (i≧0)

$\begin{matrix}{{\rho (x)} = \frac{1}{\pi \sqrt{x\left( {1 - x} \right)}}} & {{EQUATION}\quad 4}\end{matrix}$

A rational map with a parameter, such as a Katsura-Fukuda map, ageneralized Ulam-von Neumann map (equation 5), a generalized cubic map,a generalized Chebyshev map, or the like may be applied to therecurrence formula. $\begin{matrix}{{{{f\left( {l,m,x} \right)} = \frac{4{x\left( {1 - x} \right)}\left( {1 - {lx}} \right)\left( {1 - {mx}} \right)}{1 - {Ax}^{2} + {Bx}^{2} + {Cx}^{4}}}{where}A = {2\left( {l + m + {l\quad m}} \right)}}{B = {8l\quad m}}{C = {l^{2} + m^{2} - {2l\quad m} - {2l^{2}m} - {2l\quad m^{2}} + {l^{2}m^{2}}}}} & {{EQUATION}\quad 5}\end{matrix}$

For example, if the random number sequence result from the aboverecurrence formula employing the generalized Ulam-von Neumann map(equation 5), limiting distribution of the resultant random numbersequences is expressed by the following density function (equation 6)which includes the parameter of the generalized Ulam-von Neumann map.$\begin{matrix}{{\frac{1}{{K\left( {l,m} \right)}\sqrt{{x\left( {1 - x} \right)}\left( {1 - {lx}} \right)\left( {1 - {mx}} \right)}}{where}{K\left( {l,m} \right)} = {\int_{0}^{1}\frac{u}{\sqrt{\left( {1 - u^{2}} \right)\left( {1 - {lu}^{2}} \right)\left( {1 - {mu}^{2}} \right)}}}}\quad} & {{EQUATION}\quad 6}\end{matrix}$

The Katsura-Fukuda map is the same as the generalized Ulam-von Neumannmap (equation 5) where m=0.

Unexamined Japanese Patent Application KOKAI No. H10-283344 by theinventor of the present invention discloses a technique for generatingrandom numbers with utilizing rational maps. Theoretical backgrounds forthe technique are disclosed in the following documents.

S. M. Ulam and J. von Neumann, Bull. Math. Soc. 53 (1947) pp. 1120.

R. L. Adler and T. J. Rivlin, Proc. Am. Math. Soc. 15 (1964) pp. 794.

K. Umeno, Method of constructing exactly solvable chaos, Phys. Rev.E(1997) Vol. 55 pp. 5280-5284.

Conventionally, a random number sequence (a sequence of randomone-dimensional vectors) result from the above random number generatingmethod by applying a sequence of scalars (one-dimensional vectors) tothe recurrence formula as the seed.

However, such the conventional methods have the following problems.

For carrying out the Monte-Carlo method in dimension at least two, asequence of random vectors of dimensions at least two is required.However, the conventional random number generating method generates asequence of random numbers corresponding to a sequence of scalars(one-dimensional vectors). When the Monte-Carlo method is applied tosimulation in three-dimensional space, three-value selection from headof the sequence may be required for necessary times. However, suchoperation causes deviation of random number distribution, thus, errorconvergence will be poor.

Moreover, the public key encryption requires paired integers as therandom number. However, the conventional method can not performsimultaneous generation of the integers to be paired. This will make asecurity hole, thus, the encryption may be cracked.

Under such the circumstances, there is a great demand for an apparatusand method for outputting a sequence of vectors each of which comprisesplural pairs of random numbers each being generated simultaneously as amulti-dimensional random vector, and for expressing distribution of theoutput vector sequence by an analytic density function.

SUMMARY OF THE INVENTION

The present invention has been made for overcome the above problems. Itis an object of the present invention to provide to an apparatus and amethod for outputting a sequence of higher dimensional random vectors byassociating two vector sequence generating methods by which sequences ofrandom vectors whose distribution is expressed by a known analyticdensity function are output, and distribution of the resultant sequenceof random vectors is expressed by an analytic density function, and toprovide a data recording medium storing a program which realizes theabove.

To accomplish the above object, the following invention will bedisclosed in accordance with the principle of the present invention.

As shown in FIG. 1, an apparatus 100 for outputting a sequence ofvectors according to a first aspect of the present invention comprises afirst storage 101, a first calculator 102, a second storage 103, asecond calculator 104, an output 105, a first update 106, and a secondupdate 107.

(a) the first storage 101 stores vector x of dimension at least 1;

(b) the first calculator 102 calculates vector x′=f(x) which utilizes afirst rational vector map f to which the vector x stored in the firststorage 101 is input;

(c) the second storage 102 stores vector y of dimension at least 1;

(d) the second calculator 104 calculates vector y′=g(x, y) whichutilizes a second rational vector map g to which the vector x stored inthe first storage 101 and the vector y of dimension at least 1 stored inthe second storage 103 are input;

(e) the output 105 outputs vector z′ which is association of the vectorx′ resulting from the first calculator 102 and the vector y′ resultingfrom the second calculator 104;

(f) the first update 106 replaces the vector x in the first storage 101with the vector x′ which is the result of the first calculator 102; and

(g) the second update 107 replaces the vector y in the second storage103 with the vector y′ which is the result of the second calculator 104.

Here, “rational vector map” means a map which converts vector ofdimension at least 1 having rational number components into anothervector of dimension at least 1 having rational number components.

The rational vector maps f and g may be maps based on the chaos theory(described later) or arbitrary maps applicable to a recurrence formulafor random number generation. For example, a map which multiplies agiant prime number to obtain the remainder may be applicable one.

In the apparatus for outputting a sequence of vectors according to thepresent invention, a density function expressing limiting distributionof a sequence of vectors x, f(x), f(f(x)), f(f(f(x))), . . . which areobtained after applying 0 or more times the vector x of dimension atleast 1 to the first rational vector map f, is represented by ananalytic function, and

a density function expressing limiting distribution of a sequence ofvectors y, g(λ, y), g(λ, g(λ, y)), g(λ, g(λ, g(λ, y))), . . . which areobtained after applying 0 or more times the vector y of dimension atleast 1 to the second rational vector map g(λ, •) to which vector λ ofdimension at least 1 is input as a parameter, is represented by ananalytic function having the parameter λ.

As shown in FIG. 2, an apparatus 100 for outputting a sequence ofvectors according to a second aspect of the present invention comprisesa first storage 101, a first calculator 102, a second storage 103, asecond calculator 104, an output 105, a first update 106, and a secondupdate 107.

(a) the first storage 101 stores vector x of dimension at least 1;

(b) the first calculator 102 calculates vector x′=f(x) which utilizes afirst rational vector map f to which the vector x stored in the firststorage 101 is input;

(c) the second storage 101 stores vector y of dimension at least 1;

(d) the second calculator 104 calculates vector y′=g(x′, y) whichutilizes a second rational vector map g to which the vector x′ resultingthe first calculator 102 and the vector y having dimension at least 1stored in the second storage 103 are input;

(e) the output 105 outputs vector z′ which is association of the vectorx′ resulting from the first calculator 102 and the vector y′ resultingfrom the second calculator 104;

(f) the first update 106 replaces the vector x in the first storage 101with the vector x′ which is the result of the first calculator 102; and

(g) the second update 107 replaces the vector y in the second storage103 with the vector y′ which is the result of the second calculator 104.

The rational vector maps f and g may be maps based on the chaos theory(described later) or arbitrary maps applicable to a recurrence formulafor random number generation. For example, a map which multiplies agiant prime number to obtain the remainder may be applicable one.

In the apparatus for outputting a sequence of vectors according to thepresent invention, a density function expressing limiting distributionof a sequence of vectors x, f(x), f(f(x)), f(f(f(x))), . . . which areobtained after applying 0 or more times the vector x of dimension atleast one to the first rational vector map f, is represented by ananalytic function, and

a density function expressing limiting distribution of a sequence ofvectors y, g(λ, y), g(λ, g(λ, y)), g(λ, g(λ, g(λ,y))), . . . which areobtained after applying to 0 or more times the vector y of dimension atleast 1 to the second rational vector map g(λ, •) to which vector λ ofdimension at least 1 is input as a parameter, is represented by ananalytic function having the parameter λ.

The first rational vector map f for the apparatus for outputting asequence of vectors according to the present invention may be a rationalmap obtained by an addition theorem of the elliptic function,especially, one of Ulam-von Neumann map, cubic map, and quintic map, orone of Katsura-Fukuda map, generalized Ulam-von Neumann map, generalizedcubic map and generalized Chebyshev to each of which a predeterminedparameter is applied.

The second rational vector map g of the present invention may be arational map obtained by an additional theorem of the elliptic function,especially, one of Katsura-Fukuda map, generalized Ulam-von Neumann map,generalized cubic map, and generalized Chebyshev map.

In a case where the rational maps obtained by additional theorem of theelliptic function are selected as the first and second rational vectormaps f and g, a density function expressing limiting distribution of thesequence of vectors to be sequentially output by the output 105 will beobtained by a density function for a random number sequence obtained bythe selected maps.

As shown in FIG. 3, an apparatus 200 for outputting a sequence ofvectors according to a third aspect of the present invention comprises agenerator 201, a first output 202, a second output 203, and a thirdoutput 204.

(a) the generator 201 receives vector ζ of dimensions at least 2, andgenerates vector ξ of dimension at least 1 and vector η of dimension atleast 1;

(b) the first output 202 receives the vector ξ generated by thegenerator 201, and outputs vectors x[i] obtained by the followingrecurrence formula which utilizes a first rational vector map f

x[0]=ζ

x[i+1]=f(x[i]) (where i≧0);

(c) the second output 203 receives the vector η generated by thegenerator 201 and the vectors x[i] output by the first output 202, andoutputs vectors y[i] obtained by the following recurrence formula whichutilizes a second rational vector map g

y[0]=ζ

y[i+1]=g(x[i], y[i]) (where i≧0); and

(d) the third output 204 outputs vectors z[i] which is association ofthe vectors x[i] output by the first output 202 and the vectors y[i]output by the second output 203.

As shown in FIG. 4, an apparatus 200 for outputting a sequence ofvectors according to a fourth aspect of the present invention comprisesa generator 201, a first output 202, a second output 203, and a thirdoutput 204.

(a) the generator 201 receives vector ζ of dimensions at least 2, andgenerates vector ξ of dimension at least 1 and vector η of dimension atleast 1;

(b) the first output 202 receives the vector ξ generated by thegenerator 201, and outputs vectors x[i] obtained by the followingrecurrence formula which utilizes a first rational vector map f

x[0]=ζ

x[i+1]=f(x[i]) (where i≧0);

(c) the second output 203 receives the vector η generated by thegenerator 201 and the vectors x[i] output by the first output 202, andoutputs vectors y[i] obtained by the following recurrence formula whichutilizes a second rational vector map g

y[0]=η

y[i+1]=g(x[i+1], y[i]) (where i≧0); and

(d) the third output 204 outputs vectors z[i] which is association ofthe vectors x[i] output by the first output 202 and the vectors y[i]output by the second output 203.

In the apparatus for outputting a sequence of vectors according to thepresent invention, a density function expressing limiting distributionof a sequence of vectors x, f(x), f(f(x)), f(f(f(x))), . . . which areobtained after applying 0 or more times the vector x of dimension atleast one to the first rational vector map f, is represented by ananalytic function, and

a density function expressing limiting distribution of a sequence ofvectors y, g(λ, y), g(λ, g(λ, y)), g(λ, g(λ, g(λ, y))), . . . which areobtained after applying 0 or more times the vector y of dimension atleast 1 to the second rational vector map g(λ, •) to which vector λ ofdimension at least 1 is input as a parameter, is represented by ananalytic function having the parameter λ.

The first rational vector map f for the apparatus for outputting asequence of vectors according to the present invention may be a rationalmap obtained by an addition theorem of the elliptic function,especially, one of Ulam-von Neumann map, cubic map, and quintic map, orone of Katsura-Fukuda map, generalized Ulam-von Neumann map, generalizedcubic map and generalized Chebyshev to each of which a predeterminedparameter is applied.

The second rational vector map g of the present invention may be arational map obtained by an additional theorem of the elliptic function,especially, one of Katsura-Fukuda map, generalized Ulam-von Neumann map,generalized cubic map, and generalized Chebyshev map.

In these cases, an analytic density function expressing limitingdistribution of the output vector sequence is also obtained based on therational vector maps f and g.

The first output 202 itself may act as the apparatus for outputting asequence of vectors according to the present invention. This case isrealized by the following steps.

(1) Prepare an apparatus X (for outputting a sequence of vectors) whichutilizes a rational vector map f and a rational vector map g to which aparameter is input;

(2) Since the result of the apparatus X is regarded as a result of arational vector map f′, another apparatus Y for outputting a furthervector sequence is prepared by associating the apparatus X with afurther rational vector map g′ to which a parameter is input. Theapparatus Y can output a sequence of vectors whose dimension is higherthan that of the vector sequence output by the apparatus X.

(3) Repeated preparation of apparatuses in such the manner willeventually produce an apparatus for outputting a sequence of arbitrarydimensional random vectors.

A method for outputting a sequence of vectors according to a fifthaspect of the present invention comprises:

(a) the first calculation step of calculating vector x′=f(x) whichutilizes a first rational vector map f to which vector x of dimension atleast 1 stored in a first storage is input;

(b) the second calculation step of calculating vector y′=g(x, y) whichutilizes a second rational vector map g to which the vector x stored inthe first storage and vector y of dimension at least 1 stored in asecond storage are input;

(c) the output step of outputting vector z′ which is association of thevector x′ resulting from the first calculation step and the vector y′resulting from the second calculation step;

(d) the first update step of updating the first storage by storing thevector x′ obtained by the first calculation step; and

(e) the second update step of updating the second storage by storing thevector y′ obtained by the second calculation step.

A method of outputting a sequence of vectors according to a sixth aspectof the present invention comprises:

(a) the first calculation step of calculating vector x′=f(x) whichutilizes a first rational vector map f to which vector x of dimension atleast 1 stored in a first storage is input;

(b) the second calculation step of calculating vector y′=g(x′, y) whichutilizes a second rational vector map g to which the vector x′ resultingfrom the first calculation step and vector y of dimension at least 1stored in a second storage are input;

(c) the output step of outputting vector z′ which is association of thevector x′ resulting from the first calculation step and the vector y′resulting from the second calculation step;

(d) the first update step of updating the first storage by storing thevector x′ obtained by the first calculation step; and

(e) the second update step of updating the second storage by storing thevector y′ obtained by the second calculation step.

In the method for outputting a sequence of vectors according to thepresent invention, a density function expressing limiting distributionof a sequence of vectors x, f(x), f(f(x)), f(f(f(x))), . . . which areobtained after applying 0 or more times the vector x of dimension atleast one to the first rational vector map f, is represented by ananalytic function, and

a density function expressing limiting distribution of a sequence ofvectors y, g(λ, y), g(λ, g(X, y)), g(λ, g(λ, g(λ, y))), . . . which areobtained after applying 0 or more times the vector y of dimension atleast 1 to the second rational vector map g(λ, •) to which vector λ ofdimension at least 1 is input as a parameter, is expressed by ananalytic function having the parameter λ.

The first rational vector map f for the method for outputting a sequenceof vectors according to the present invention may be a rational mapobtained by an addition theorem of the elliptic function, especially,one of Ulam-von Neumann map, cubic map, and quintic map, or one ofKatsura-Fukuda map, generalized Ulam-von Neumann map, generalized cubicmap and generalized Chebyshev to each of which a predetermined parameteris applied.

The second rational vector map g of the method for outputting a sequenceof vectors according to the present invention may be a rational mapobtained by an additional theorem of the elliptic function, especially,one of Katsura-Fukuda map, generalized Ulam-von Neumann map, generalizedcubic map, and generalized Chebyshev map.

In this case, an analytic density function expressing distribution ofthe output vector sequence is also obtained based on the rational vectormaps f and g.

A program which realizes the apparatus and method for outputting asequence of vectors according to the present invention may be stored ina data recording medium such as a compact disc, a floppy disk, a harddisk, a magneto-optical disk, a digital versatile (video) disk, amagnetic tape, and a semiconductor memory apparatus.

The program may be distributed by a carrier wave signal.

A data processor having a storage, a calculator, an output apparatus,and the like, for example, a general purpose computer, a video gameapparatus, a PDA (Personal Data Assistance), a mobile phone, and thelike acts as the apparatus for outputting a sequence of vectors or forrealizing the method, by executing the program stored in the datarecording medium according to the present invention.

The data recording medium storing the program according to the presentinvention may be distributed or merchandised separately from the dataprocessor apparatus.

BRIEF DESCRIPTION OF THE DRAWINGS

These objects and other objects and advantages of the present inventionwill become more apparent upon reading of the following detaileddescription and the accompanying drawings in which:

FIG. 1 is a block diagram showing the structure of an apparatus foroutputting a sequence of vectors according to a first aspect of thepresent invention;

FIG. 2 is a block diagram showing the structure of an apparatus foroutputting a sequence of vectors according to a second aspect of thepresent invention;

FIG. 3 is a block diagram showing the structure of an apparatus foroutputting a sequence of vectors according to a third aspect of thepresent invention;

FIG. 4 is a block diagram showing the structure of an apparatus foroutputting a sequence of vectors according to a fourth aspect of thepresent invention;

FIG. 5 is a block diagram showing the structure of a data processorwhich acts as the apparatus for outputting a sequence of vectorsaccording to the present invention;

FIG. 6 is a flowchart showing the steps executed by the apparatus foroutputting a sequence of vectors according to the present invention;

FIG. 7 is a graph showing density function of random number distributionwhen the generalized Ulam-von Neumann map is applied to the recurrenceformula;

FIG. 8 is a diagram showing plotted coordinates represented bytwo-dimensional vectors output at step S409 in the flowchart shown inFIG. 6;

FIG. 9 is a histogram whose coordinates are the two-dimensional vectorsoutput at step S409 in the flowchart shown in FIG. 6; and

FIG. 10 is a graph representing a cross section of the histogram shownin FIG. 6.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS First Embodiment

FIG. 5 is a block diagram showing the structure of a data processor suchas a general purpose computer which acts as an apparatus for outputtinga sequence of vectors according to an embodiment of the presentinvention.

A data processor 301 is under control of a CPU (Central Processing Unit)302. A main storage 303 such as a RAM (Random Access Memory) storestemporary data, and an external storage such as a hard disk, a floppydisk, a CD-ROM (Compact Disc Read Only Memory), a magnetic tape and amagneto-optical disk stores programs to be executed by the CPU 302.

Upon the data processor 301 is turned on, the CPU 302 executes a bootprogram stored in a ROM (Read Only Memory) 308. Upon the boot programexecution, programs such as OS (Operating System) and variousapplication programs stored in the external storage 304, etc. are loadedto the main storage 303, and the CPU 302 executes them.

Files including results of the execution will be stored in the externalstorage 304, or the results will be displayed on a display 305 such as aCRT (Cathode Ray Tube) display, and a liquid crystal display. A userinputs commands to the data processor 301 through an input device 306such as a mouse and a keyboard.

In a case where the data processor 301 acts as the apparatus 100 foroutputting a sequence of vectors shown in FIGS. 1 and 2, the mainstorage 303 acts as the first and second storage 101 and 103; the CPU302 acts as the first calculator 102, the second calculator 104, thefirst update 106, and the second update 107; the external storage 304acts as the output 105 when saving the result files; the display 305acts as the output 105 when displaying the results; and the main storage303 acts as the output 105 when sharing the results with other programs.

In another case where the data processor acts as the apparatus 200 foroutputting a sequence of vectors shown in FIG. 3, the CPU 302collaborates with the main storage 303 (the external storage 304 and thedisplay 305 as needed) to act as the generator 201, the first output202, the second output 203, and the third output 204.

The main storage 303 and the external storage 304 act as the datarecording medium according to the present invention. The ROM 308 may actas the data recording medium according to the present invention.

A process flow executed by the apparatus for outputting a sequence ofvectors according to the present invention will now be described withreference to FIG. 6. FIG. 6 is a flowchart showing the process flow ofthe present invention.

For the sake of simple explanation, Ulam-von Neumann map (seeequation 1) and Katsura-Fukuda map are exemplified hereinafter as therational map f and the rational map g respectively. However, thoseskilled in the art may easily modify the embodiment so as to employother maps, and such the modified embodiments are included in the scopeof the present invention.

The CPU 302 obtains a time value or the like as a seed for the randomnumber (step S401). In this case, a sequence of scalars (1-dimensionalvectors) is input to the rational map f, and another sequence of scalars(1-dimensional vectors) and a parameter are input to the rational map g.Therefore, two different seeds for the scalars are required. In otherwords, the seeds for the random number are 2-dimensional vectors.

The user may input the seeds for the random number through the inputdevice 306. And the input seeds may be associated with the time value.This method is helpful for generating a public key for encryption.

The CPU 302 transfers the seeds to the first storage 101 and the secondstorage 103 respectively (step S402). Thus, an initial value for therandom number generation is set.

Steps S401-S402 correspond to process executed by the generator 201 inthe apparatus 200 for outputting a sequence of random vectors shown inFIG. 3.

Then the CPU 302 fetches a value x stored in the first storage 101 and avalue y stored in the second storage 103 (step S403), and calculatesy′=g(x, y) (step S404). The CPU 302 updates the second storage 103 bystoring the resultant value y′ (step S405).

That is, the CPU 302 acts as the second calculator 104 in step S404.

The CPU 302 further fetches the value x stored in the first storage 101(step S406), and calculates x′=f(x) (step S407). The CPU 302 updates thefirst storage 101 by storing the resultant value x′ (step S408).

That is, the CPU 302 acts as the first calculator 102 in step S407.

Finally, the CPU 302 associates the value x′ in the first storage 101with the value y′ in the second storage 103, and outputs a resultantvalue to the external storage 304 or the like (step S409). Then, theflow returns to step S403.

A result after associating n-dimensional vector with anotherm-dimensional vector is (n+m)-dimensional vector. Element of the resultincludes laid elements of the n-dimensional vector, and laid elements ofthe m-dimensional vector follows thereto. Process executed by thegenerator 201 in the apparatus 200 for outputting a sequence of vectorsshown in FIG. 3 is realized by disassociating the vectors.

Random (m+n)-dimensional vectors will be output to the external storage304 after repeat execution of the above steps.

The relationships between process executed by the first to third outputs202-204 in the apparatus 200 for outputting a sequence of vectors shownin FIG. 3 and steps shown in FIG. 6 are as follows:

the first output 202 executes steps S406 to S408; the second output 203executes steps S403 to S405; and the third output 204 executes stepS406.

As mentioned above, the generalized Ulam-von Neumann map (seeequation 1) is employed as the rational vector map f in this embodiment.FIG. 7 shows a result of a density function expressing random numberdistribution where the generalized Ulam-von Neumann map is employed as arecurrence formula. As shown in FIG. 7, the density function showsununiformed result in the range of 0≦x≦1 (concave up where 0<x<1;infinite where x=0 and x=1).

As mentioned above, the Katsura-Fukuda function (equation 7) is employedas the rational vector map g in this embodiment. The random numberdistribution is analytically expressed by equation 8 when theKatsura-Fukuda function is employed as a recurrence formula where theparameter x′ is fixed. $\begin{matrix}{{g\left( {x^{\prime},y} \right)} = \frac{4{y\left( {1 - y} \right)}\left( {1 - {x^{\prime}y}} \right)}{\left( {1 - {x^{\prime}y^{2}}} \right)^{2}}} & {{EQUATION}\quad 7} \\{{{{\nu \left( {x,y} \right)} = \frac{1}{{K(x)}\sqrt{{y\left( {1 - y} \right)}\left( {1 - {xy}} \right)}}}{where}{K(x)} = {\int_{0}^{1}\frac{u}{\sqrt{\left( {1 - u^{2}} \right)\left( {1 - {x\quad u^{2}}} \right)}}}}\quad} & {{EQUATION}\quad 8}\end{matrix}$

FIG. 8 is a diagram showing sequentially plotted dots whose coordinationis represented by 2-dimensional vectors output at step S409. FIG. 9 is ahistogram whose coordination is represented by the 2-dimensional vectorsoutput at step S409. FIG. 10 shows a cross section of the histogramshown in FIG. 9.

The inventor mathematically proved the following fact:

“Generally, when a rational ergodic map f having a density function p(x)and a rational ergodic map g(x, •) having a parameter x and a densityfunction v(x, y) are associated with each other by the method of thepresent invention which utilizes a recurrence formula, a densityfunction expressing limiting distribution of the output vector sequencewill be v(x, y)ρ(x).”

That is, if known analytic density functions expressing limitingdistribution are associated with each other, the resultant densityfunction of limiting distribution will be a product of the originaldensity functions. Therefore, the resultant density function is revanalytically available.

The above fact is also proved by the graph shown in FIG. 10 because itshows the shape which is almost the same as the shape of the graph shownin FIG. 7. Moreover, the method of the present invention realizes lessdeflection of random numbers than that caused by the conventionaltechniques.

Modified order of the process flow, parallel execution of separatedprocess flow, or the like may realize the same effect of the presentinvention. Such the modification is included in the scope of theinvention.

Second Embodiment

In the first embodiment, the apparatus for outputting a sequence ofvectors was the data processor such as a general purpose computer. Anelectronic circuit which acts as the apparatus for outputting a sequenceof vectors will be described in a second embodiment.

Flip-flop type storage circuits act as the first storage 101 and thesecond storage 103 in the apparatus 100 for outputting a sequence ofvectors shown in FIG. 1.

A combination of an addition circuit and a multiplication circuit actsas the first and second calculators 102 and 104.

Output lines of the circuit acting as the second calculator 104 act asthe output 105.

To realize function of the first and second updates 106 and 107,constantly delaying clock of signals output from the output lines of thecircuit acting as the first and second calculators 102 and 104, andstoring the delayed outputs in the first and second storage 101 and 103as feedback.

Such the circuit based device for outputting a sequence of vectorsaccording to the present invention will be helpful to apply the presentinvention to a device such as a PDA and a mobile phone which requiressimple and small structure with less power consumption.

Third Embodiment

In the above embodiments, rational maps obtained by addition theorem forthe elliptic function have been applied. Some of other maps obtained byelliptic integral, hyperelliptic integral or their modifications havethe similar characteristics, therefore, such the maps may be utilized.Or, another map which represents a recurrence formula for theconventional random number generating technique may be utilized.

Fourth Embodiment

An inverse function technique established by von Neumann may generates asequence of uniformly distributed random vectors based on the sequenceof random vectors (regardless of their dimensions) obtained by thepresent invention, because the density function expressing the sequenceof random vectors is analytically available.

Fifth Embodiment

Another available technique which is helpful to realize the presentinvention is data streaming. The data streaming is a major technique forOS such as UNIX, logical languages such as prolog and GHC, functionallanguages such as Lisp and Haskell, and the like. More precisely, thepresent invention will be realized after preparing the following datastreaming processes A and B regarding to the rational vector maps f andg.

The process A generates streamed data for a sequence of vectors x[i](process A is expressed as Predicate, Function, Procedure, or the likein the programming language).

Process A:

 x[0], x[1], x[2], x[3], . . .

x[0]=ζ

x[i+1]=f(x[i]) (where i≧0)

And the following process B generates streamed data for a sequence ofvectors y[i] after receiving the streamed data x[i] output by process A.

Process B:

y[0], y[1], y[2], y[3], . . .

y[0]=η

y[i+1]=g(x[i], y[i]) (where i≧0)

Communication between process A and process B may be described byso-called Producer-Consumer model, and known techniques such asgeneration on demand, and streamed data buffering may be applicable tothe communication technique.

Sixth Embodiment

In the above embodiments, resultant values after repeated calculationswere used, that is, the following calculation was carried out for a casewhere the rational vector map g is utilized.

y′=g(x, y)

y[i+1]=g(x[i], y[i])

In this embodiment, the following recurrence formula will be employedinstead of the above.

y′=g(x′, y)

y[i+1]=g(x[i+1], y[i])

Effects brought by this embodiment are the same as those by the aboveembodiments.

As described above, the present invention brings the following effects.

The present invention provides an apparatus and a method for outputtinga sequence of higher dimensional vectors whose limiting distribution isexpressed by a known analytic density function, by associating twovector sequence generating methods by which sequences of random vectorswhose limiting distribution is expressed by a known analytic functionare output.

Moreover, the present invention realizes easy observation ofdistribution characteristics, because the density function for limitingdistribution of the resultant sequence of higher dimensional randomvectors is represented by a product of the density functions expressinglimiting distribution of a vector sequence generated by the knownmethods. This effect may be applicable to various fields. In a casewhere rational maps are utilized as the known generation methods, allelements in the resultant sequence of vectors will be rational numbers.This effect is helpful to strictly keep calculation accuracy of acomputer.

The sequence of random vectors generated by the present invention may beapplicable to Monte-Carlo method, CDMA technology for mobile or opticalcommunications, public key encryption for telecommunications viainternet etc, and the like.

Furthermore, a data recording medium storing a program which realizesthe present invention may be merchandized or distributed as a softwarepackage separated from a data processor. The program stored in the datarecording medium according to the present invention will be executed bya data processor such as a general purpose computer in order to realizethe apparatus and the method for outputting a sequence of vectorsaccording to the present invention.

Various embodiments and changes may be made thereunto without departingfrom the broad spirit and scope of the invention. The above-describedembodiments are intended to illustrate the present invention, not tolimit the scope of the present invention. The scope of the presentinvention is shown by the attached claims rather than the embodiments.Various modifications made within the meaning of an equivalent of theclaims of the invention and within the claims are to be regarded to bein the scope of the present invention.

This application is based on Japanese Patent Application Nos. H11-85744filed on Mar. 29, 1999 and H11-362203 filed on Dec. 21, 1999, andincluding specification, claims, drawings and summary. The disclosure ofthe above Japanese Patent Application is incorporated herein byreference in its entirety.

What is claimed is:
 1. An apparatus for outputting a sequence of vectorscomprising: (a) a first storage which stores vector x of dimension atleast 1; (b) a first calculator which calculates vector x′=f(x) whichutilizes a first rational vector map f to which the vector x stored insaid first storage is input; (c) a second storage which stores vector yof dimension at least 1; (d) a second calculator which calculates vectory′=g(x′, y) which utilizes a second rational vector map to which thevector x′ resulting from said first calculator and the vector y ofdimension at least 1 stored in said second storage are input; (e) anoutput which outputs z′ which is association of the vector x′ resultingfrom said first calculator and the vector y′ resulting from said secondcalculator; (f) a first update which updates said first storage bystoring the vector x′ resulting from said first calculator; and (g) asecond update which updates said second storage by storing the vector y′resulting from said second calculator, where the rational vector map isa map which converts vector of dimension at least 1 having rationalnumber components into another vector of dimension at least 1 havingrational number components.
 2. The apparatus according to claim 1,wherein a density function which expresses limiting distribution of asequence of vectors x, f(x), f(f(x)), f(f(f(x))), . . . which areobtained after applying 0 or more times the vector x of dimension atleast 1 to the first rational vector map f, is represented by ananalytic function, and a density function expressing limitingdistribution of a sequence of vectors y, g(λ, y), g(λ, g(λ, y)), g(λ,g(λ, g(λ, y))), . . . which are obtained after applying 0 or more timesthe vector y of dimension at least 1 to the second rational vector mapg(λ, •) to which vector λ of dimension at least 1 is input as aparameter, is represented by an analytic function having the parameterλ.
 3. The apparatus according to claim 2, wherein the first rationalvector map f is obtained by an addition theorem of the ellipticfunction, especially, one of Ulam-von Neumann map, cubic map, andquintic map, or one of Katsura-Fukuda map, generalized Ulam-von Neumannmap, generalized cubic map and generalized Chebyshev to each of which apredetermined parameter is applied.
 4. The apparatus according to claim2, wherein the second rational vector map g is obtained by an additionaltheorem of the elliptic function, especially, one of Katsura-Fukuda map,generalized Ulam-von Neumann map, generalized cubic map, and generalizedChebyshev map.
 5. An apparatus for outputting a sequence of vectorscomprising: (a) a generator which receives vector ζ of dimensions atleast 2, and generates vector ξ of dimension at least 1 and vector η ofdimension at least 1; (b) a first output which outputs vectors x[i]obtained by the vector ζ generated by said generator and the followingrecurrence formula which utilizes a first rational vector map f x[0]=ζx[i+1]=f(x[i]) (where i≧0); (c) a second output which outputs vectorsy[i] obtained by the vectors x[i] generated by said first output and thefollowing recurrence formula which utilizes a second rational vector mapg y[0]=η  y[i+1]=g(x[i+1], y[i]) (where i≧0); and (d) a third outputwhich outputs vectors z[i] which is association of the vectors x[i]output by said first output and the vectors y[i] output by said secondoutput.
 6. The apparatus according to claim 5, wherein a densityfunction expressing limiting distribution of a sequence of vectors x,f(x), f(f(x)), f(f(f(x))), . . . which are obtained after applying 0 ormore times the vector x of dimension at least one to the first rationalvector map f, is represented by an analytic function; and a densityfunction expressing limiting distribution of a sequence of vectors y,g(λ, y), g(λ, gλ, y)), g(λ, g(λ, g(λ, y))), . . . which are obtainedafter applying 0 or more times the vector y of dimension at least 1 tothe second rational vector map g(λ, •) to which vector λ of dimension atleast 1 is input as a parameter, is represented by an analytic functionhaving the parameters λ.
 7. The apparatus according to claim 6, whereinthe first rational vector map f is obtained by an addition theorem ofthe elliptic function, especially, one of Ulam-von Neumann map, cubicmap, and quintic map, or one of Katsura-Fukuda map, generalized Ulam-vonNeumann map, generalized cubic map and generalized Chebyshev to each ofwhich a predetermined parameter is applied.
 8. The apparatus accordingto claim 6, wherein the second rational vector map g is obtained by anadditional theorem of the elliptic function, especially, one of KatsuraFukuda map, generalized Ulam-von Neumann map, generalized cubic map, andgeneralized Chebyshev map.
 9. A carrier wave in which has been embeddeda signal representing a program which executes: (a) the firstcalculation step of calculating vector x′=f(x) which utilizes a firstrational vector map f to which vector x of dimension at least 1 storedin a first storage; (b) the second calculation step of calculatingvector y′=g(x′, y) which utilizes a second rational vector map g towhich the vector x′ resulting from said first calculation step andvector y of dimension at least 1 stored in a second storage are input;(c) the output step of outputting vector I which is association of thevector x′ resulting from said first calculation step and the vector y′resulting from said second calculation step; (d) the first update stepof updating said first storage by storing the vector x′ resulting fromsaid first calculation step; and (e) the second update step of updatingsaid second storage by storing the vector yt resulting from said secondcalculation step.
 10. The carrier wave according to claim 9, wherein adensity function which expresses limiting distribution of a sequence ofvectors x, f(x), f(f(x)), f(f(f(x))), . . . which are obtained afterapplying 0 or more times the vector x of dimension at least 1 to thefirst rational vector map f, is represented by an analytic function, anda density function expressing limiting distribution of a sequence ofvectors y, g(X, y), g(λ, g(λ, y)), g(λ, g(λ, g(λ, y))), . . . which areobtained after applying 0 or more times the vector y of dimension atleast 1 to the second rational vector map g(λ, •) to which vector λ ofdimension at least 1 is input as a parameter, is represented by ananalytic function having the parameter λ.
 11. The carrier wave accordingto claim 10, wherein the first rational vector map f is obtained by anaddition theorem of the elliptic function, especially, one of Ulam-vonNeumann map, cubic map, and quintic map, or one of Katsura-Fukuda map,generalized Ulam-von Neumann map, generalized cubic map and generalizedChebyshev to each of which a predetermined parameter is applied.
 12. Thecarrier wave according to claim 10, wherein the second rational vectormap g is obtained by an additional theorem of the elliptic function,especially, one of Katsura-Fukuda map, generalized Ulam-von Neumann map,generalized cubic map and generalized Chebyshev map.